Factor copula models for mixed data

We develop factor copula models to analyse the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and nonlinear dependence. They can be explained as conditional independence models with latent variables that do not necessarily have an additive latent structure. We focus on important issues of interest to the social data analyst, such as model selection and goodness of fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by reanalysing three mixed response data sets. Our studies suggest that there can be a substantial improvement over the standard factor model for mixed data and make the argument for moving to factor copula models.     

Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling

We relate Thurstonian models for paired comparisons data to Thurstonian models for ranking data, which assign zero probabilities to all intransitive patterns. We also propose an intermediate model for paired comparisons data that assigns nonzero probabilities to all transitive patterns and to some but not all intransitive patterns.There is a close correspondence between the multidimensional normal ogive model employed in educational testing and Thurstone's model for paired comparisons data under multiple judgment sampling with minimal identification restrictions. Alike the normal ogive model, Thurstonian models have two formulations, a factor analytic and an IRT formulation. We use the factor analytic formulation to estimate this model from the first and second order marginals of the contingency table using estimators proposed by Muthén. We also propose a statistic to assess the fit of these models to the first and second order marginals of the contingency table. This is important, as a model may reproduce well the estimated thresholds and tetrachoric correlations, yet fail to reproduce the marginals of the contingency table if the assumption of multivariate normality is incorrect.A simulation study is performed to investigate the performance of three alternative limited information estimators which differ in the procedure used in their final stage: unweighted least squares (ULS), diagonally weighted least squares (DWLS), and full weighted least squares (WLS). Both the ULS and DWLS show a good performance with medium size problems and small samples, with a slight better performance of the ULS estimator.This paper is based on the author's doctoral dissertation; Ulf Böckenholt, advisor. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. The author is indebted to Adolfo Hernández for stimulating discussions that helped improve this paper, and to Ulf Böckenholt and the Associate Editor for a number of helpfulsuggestions to a previous draft.     

Estimation for structural equation models with missing data

A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.This research was supported in part by Research Grant DAD1070 from the U.S. Public Health Service. The author is indebted to anonymous reviewers for some very valuable suggestions. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

认知诊断模型资料拟合检验方法和统计量

认知诊断模型界定了测验题目和所考察属性之间的关系, 通过被试的作答反应获取被试对属性或知识技能的掌握情况。认知诊断模型资料拟合检验可以从项目拟合、模型绝对拟合、模型相对拟合和个人拟合方等方面进行。通过对认知诊断拟合检验方法和统计量的详细介绍和评价, 可为认知诊断实践提供借鉴和参考。未来研究可在更丰富的研究条件下对各统计量的性能进行评价和对比, 完善已有的拟合检验方法, 提出新的拟合统计量。     

两种新的计算机化自适应测验在线标定方法

在线标定技术由于具有诸多优点而被广泛应用于计算机化自适应测验(CAT)的新题标定。Method A是想法最直接、算法最简单的CAT在线标定方法, 但它具有明显的理论缺陷--在标定过程中将能力估计值视为能力真值。将全功能极大似然估计方法(FFMLE)与“利用充分性结果”估计方法(ECSE)的误差校正思路融入Method A (新方法分别记为FFMLE-Method A和ECSE-Method A), 从理论上对能力估计误差进行校正, 进而克服Method A的标定缺陷。模拟研究的结果表明：(1)在大多数实验条件下, 两种新方法较Method A总体上可以改进标定精度, 且在测验长度为10的短测验上的改进幅度最大; (2)当CAT测验长度较短或中等(10或20题)时, 两种新方法的表现与性能最优的MEM已非常接近。当测验长度较长(30题)时, ECSE-Method A的总体表现最好、优于MEM; (3)样本量越大, 各种方法的标定精度越高。     

Exact tests for the rasch model via sequential importance sampling

Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.This Research was supported in part by National Science Foundation grant DMS-0203762 and a University of Pennsylvania Research Foundation grant.The authors are grateful to Don Burdick for helpful comments. In addition, the authors wish to thank the editor, the associate editor, and the referees for their helpful suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Limited‐information goodness‐of‐fit testing of diagnostic classification item response models

Despite the growing popularity of diagnostic classification models (e.g., Rupp et al., 2010, Diagnostic measurement: theory, methods, and applications, Guilford Press, New York, NY) in educational and psychological measurement, methods for testing their absolute goodness of fit to real data remain relatively underdeveloped. For tests of reasonable length and for realistic sample size, full‐information test statistics such as Pearson's X² and the likelihood ratio statistic G² suffer from sparseness in the underlying contingency table from which they are computed. Recently, limited‐information fit statistics such as Maydeu‐Olivares and Joe's (2006, Psychometrika, 71, 713) M₂ have been found to be quite useful in testing the overall goodness of fit of item response theory models. In this study, we applied Maydeu‐Olivares and Joe's (2006, Psychometrika, 71, 713) M₂ statistic to diagnostic classification models. Through a series of simulation studies, we found that M₂ is well calibrated across a wide range of diagnostic model structures and was sensitive to certain misspecifications of the item model (e.g., fitting disjunctive models to data generated according to a conjunctive model), errors in the Q‐matrix (adding or omitting paths, omitting a latent variable), and violations of local item independence due to unmodelled testlet effects. On the other hand, M₂ was largely insensitive to misspecifications in the distribution of higher‐order latent dimensions and to the specification of an extraneous attribute. To complement the analyses of the overall model goodness of fit using M₂, we investigated the utility of the Chen and Thissen (1997, J. Educ. Behav. Stat., 22, 265) local dependence statistic $X_{\mathrm{LD}}^2$ for characterizing sources of misfit, an important aspect of model appraisal often overlooked in favour of overall statements. The $X_{\mathrm{LD}}^2$ statistic was found to be slightly conservative (with Type I error rates consistently below the nominal level) but still useful in pinpointing the sources of misfit. Patterns of local dependence arising due to specific model misspecifications are illustrated. Finally, we used the M₂ and $X_{\mathrm{LD}}^2$ statistics to evaluate a diagnostic model fit to data from the Trends in Mathematics and Science Study, drawing upon analyses previously conducted by Lee et al., (2011, IJT, 11, 144).     

Constructing the exact significance level for a person fit statistic

This paper uses an extension of the network algorithm originally introduced by Mehta and Patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the Rasch model. By assuming that item difficulties are known, the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score. A simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less.     

Loglinear Rasch models for the analysis of stability and change

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.The author is greatly indebted to Ulf Böckenholt, Rolf Langeheine, and several anonymous reviewers for many helpful suggestions.     

Goodness of fit tests for paired comparison models

A goodness of fit test presented by Andersen is shown to be incorrect. The correct test is described and a re-analysis of Andersen's data is provided.     

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.This research was supported in part by a research grant DA01070 from the U.S. Public Health Service. The production assistance of Julie Speckart is gratefully acknowledged.     

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

Researchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.     

Pooling data versus averaging model fits for some prototypical multinomial processing tree models

For a study with multinomial data where there are n_g individuals and with each person having n_r test trials, the question arises as to how to fit the parameters of a multinomial processing tree (MPT) model. Should each parameter be estimated for each individual and then averaged to obtain a group estimate, or should the frequencies in the multinomial categories be pooled so that the model is fit once for the entire group? This basic question is explored with a series of Monte Carlo simulations for some prototypical MPT models. There is a general finding of a pooling advantage for the case where there is a single experimental condition. Also when there are different experimental conditions, there is reduced bias for detecting condition differences for a method based on the pooled data. Although the focus of the paper is on multinomial models, a general theorem is advanced that establishes a basic condition that determines whether there is or is not a difference between the averaging of individual estimates and the estimate based on the pooled data.     

Nonunique solutions to the likelihood equation for the three-parameter logistic model

Samejima identified the possibility of multiple solutions to the likelihood equation (multiple maxima in the likelihood function) for estimating an examinee's trait value for the three-parameter logistic model. In the practical applications that Lord studied, he found that multiple solutions did not occur when the number of items was 20. In the present paper, fourteen multiple-choice achievement tests with from 20 to 50 items were examined to see if it was possible for them to produce item response vectors with multiple maxima; such vectors were found for all the tests. Examination of response vectors for large groups of real examinees found that from 0 to 3.1% of them had response vectors with multiple maxima. The implications of these results for multiple-choice tests are discussed.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

Paired comparisons with individual differences

A model of Bradley-Terry type for paired comparisons is considered. In addition to the usual parameters, the model allows for individual parameters and corresponding parameters for the choice-alternatives. The model is applied to a set of data from a Danish investigation of the attitude of blue collar workers towards alternative social gains. The proposed model is finally compared to a model recently suggested by Schönemann and Wang.     

Statistical consistency and hypothesis testing for nonmetric multidimensional scaling

The properties of nonmetric multidimensional scaling (NMDS) are explored by specifying statistical models, proving statistical consistency, and developing hypothesis testing procedures. Statistical models with errors in the dependent and independent variables are described for quantitative and qualitative data. For these models, statistical consistency often depends crucially upon how error enters the model and how data are collected and summarized (e.g., by means, medians, or rank statistics). A maximum likelihood estimator for NMDS is developed, and its relationship to the standard Shepard-Kruskal estimation method is described. This maximum likelihood framework is used to develop a method for testing the overall fit of the model.     

Analysis of Interactions and Nonlinear Effects with Missing Data: A Factored Regression Modeling Approach Using Maximum Likelihood Estimation

Abstract

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a multivariate normal distribution, which is also the default in many statistical software packages. This distribution will in general be misspecified if predictors with missing data have nonlinear effects (e.g., x²) or are included in interaction terms (e.g., x·z). In the present article, we introduce a factored regression modeling approach for estimating regression models with missing data that is based on maximum likelihood estimation. In this approach, the model likelihood is factorized into a part that is due to the model of interest and a part that is due to the model for the incomplete predictors. In three simulation studies, we showed that the factored regression modeling approach produced valid estimates of interaction and nonlinear effects in regression models with missing values on categorical or continuous predictor variables under a broad range of conditions. We developed the R package mdmb, which facilitates a user-friendly application of the factored regression modeling approach, and present a real-data example that illustrates the flexibility of the software.     

缺失数据的结构方程建模:全息极大似然估计时辅助变量的作用

本研究通过蒙特卡洛模拟考查了采用全息极大似然估计进行缺失数据建模时辅助变量的作用。具体考查了辅助变量与研究变量的共缺机制、共缺率、相关程度、辅助变量数目与样本量等因素对参数估计结果精确性的影响。结果表明,当辅助与研究变量共缺时:(1)对于完全随机缺失的辅助变量,结果更容易出现偏差;(2)对于MAR-MAR组合机制,纳入单个辅助变量是有益的;对于MAR-MCAR或MAR-MNAR组合机制,纳入多于一个辅助变量的效果更好;(3)纳入与研究变量低相关的辅助变量对结果也是有益的。     

On testing the equality of proportions in the paired case with incomplete data

The problem of testing two correlated proportions with incomplete data is considered by means of Monte Carlo simulations studies. A test proposed in this paper, which can be regarded as a generalization of McNemar's test, is recommended in all cases with incomplete data and not too small samples.